An alternative deduction of the Hill-type surfaces of the spatial 3-body problem. (English) Zbl 0757.70005
Summary: Inequalities stronger than Sundman’s and the best possible zero velocity surfaces of the spatial three-body problem first obtained by D. G. Saari [Celestial Mech. 40, 197-223 (1987; Zbl 0636.70008)] are deduced using a modified version of the transformation developed by K. Zare [ibid. 14, 73-83 (1976; Zbl 0349.70005)]. The notion of inertia ellipsoid is used to show the equivalence of the present authors’ result to that of Saari’s.
MSC:
| 70F07 | Three-body problems |
| 70F15 | Celestial mechanics |
| 70H15 | Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics |
References:
| [1] | Golubev V.G.: 1968, ?Hill Stability in the Unrestricted Three-Body Problem?, Soviet Phys. Doklady 13, 373-375. · Zbl 0187.22301 |
| [2] | Lundberg, J. and Szebehely, V. et al.: 1985, Celest. Mech. 36, 191. · Zbl 0572.70011 · doi:10.1007/BF01230651 |
| [3] | Poincare, H.: 1892, Les Methods Nouvelles de la Mechanique Celeste, Gauthier Villars. |
| [4] | Saari, D.G.: 1976, ?The N-Body Problem of Celestical Mechanics?, Celest. Mech. 14, 11-17. · Zbl 0345.70005 · doi:10.1007/BF01247127 |
| [5] | Saari, D.G.: 1987, ?From Rotations and Inclinations to Zero Configurational Velocity Surfaces II. The Best Possible Configurational Velocity Surfaces?, Celest. Mech. 40, 197-223. · Zbl 0636.70008 · doi:10.1007/BF01235841 |
| [6] | Whittaker, E.T.: 1904, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press. · JFM 35.0682.01 |
| [7] | Wintner, A.: 1910, 1947, The Analytical Foundations of Celestial Mechanics, Princeton University Press. · JFM 67.0785.01 |
| [8] | Zare, K.: 1976, ?The Effects of Integrals on the Totality of Solutions of Dynamical Systems?, Celest. Mech. 14, 73-83. · Zbl 0349.70005 · doi:10.1007/BF01247133 |
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